Question

If $S_1,S_2,.S_n$ are the sums of infinite geometric series whose first terms are $1,2,\mathrm{3..}n$ and common ratio are $\frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{n+1}$ respectively then prove that
$S_1+S_2+S_3+...+S_n=\frac{1}{2}n(n+3).$

Answer 1

$S_n=$ sum of an infinite $G.P.$ whose first term is $n$ and common ration $r=1/(n+1)$
$S_n=\frac{a}{1-r}=\frac{n}{1-\left[1/(n+1)\right]}=n+1$.
Putting $n=1,2,3,..,n$
$S_1+S_2+S_3+..+S_n=2+3+4++n+1$
$=\left(n/2\right)[2.2+(n-1\left).1\right]=\frac{1}{2}n(n+3)$.